Does the Earth have more than two magnetic poles?

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Yes, but only two of them are dramatically obvious.

Any physical system in which a particular quantity ( temperature, density, force, speed) is distributed in space, can be mathematically represented in terms of a combination of harmonic functions. For systems such as spherical planets or stars, the mathematical functions you use are called 'Spherical Legendre Polynomials'. The lowest-order term is just a constant intensity or magnitude of some physical quantity, distributed over the surface of a perfect sphere with the same radius as your planet or star. The 'higher-order' terms represent dipolar ( 2-poles), quadrupolar (4-poles) and even higher 'multipolar' distributions.

The magnetic field of the earth, as any grade school students will tell you, has a magnetic north and south pole. In terms of the previous paragraph, this means that the multipolar decomposition of the field has a dominant 'dipolar' term. This is expected from basic physics because magnetic fields are intrinsically dipolar. BUT, if the cause of these magnetic fields is in more complicated currents of particles deep inside the earth, then the dipolar term will not be enough to describe the complete field. This means that, in this mathematical exercise, the higher-order terms will not be zero. In a dipolar field, you have two opposite-polarity quantities ( north and south magnetic poles) located on diametrically opposite geographic hemispheres. For the earth, this is not exactly the case because the north and south magnetic poles are not at geographically opposite sides of the earth. In fact, a line drawn between the actual poles passes about 500-700 miles away from the geometric center of the earth. 'Offset dipole magnetic fields' are the rule, not the exception in the other planets in our solar system. Why this happens is not known.

Anyway, the earth has two dominant magnetic poles, and several very weak 'quadrupolar' poles of which there are, at least mathematically, about 8 in number. These poles are far weaker than the dipole field, and measure only weak departures of the local geographic field strength from the basic dipolar North-South field.